Drinfeld Isomorphism for Novel Quantum Affine Algebra of Type $A_{1}^{(1)}$

In this paper, we first review the definition of the novel quantum affine algebra (U_{\textbf{q}}(\widehat{\mathfrak{sl}}2)) of type (A{1}^{(1)}) given in \cite{FHZ, HZhuang}. Furthermore, by introducing (Ω)-invariant generating functions, we construct the Drinfeld realization (U^{D}_{\textbf{q}}(\widehat{\mathfrak{sl}}2)) of this algebra, and prove that (U{\textbf{q}}(\widehat{\mathfrak{sl}}2)) and (U^{D}{\textbf{q}}(\widehat{\mathfrak{sl}}_2)) are algebraically isomorphic, which is known as the Drinfeld Isomorphism.

💡 Research Summary

In this paper the authors investigate a two‑parameter quantum affine algebra of type (A_{1}^{(1)}), denoted (U_{\mathbf q}(\widehat{\mathfrak{sl}}{2})), and establish its Drinfeld realization together with an explicit Drinfeld isomorphism. The work builds on earlier studies that introduced a “novel” quantum affine algebra with parameters (\epsilon{1}=\epsilon_{2}=-1) and (t=1). Unlike the standard one‑parameter quantum affine algebra, this algebra possesses a non‑symmetric set of defining relations and is known not to be isomorphic to the usual Drinfeld–Jimbo version.

The paper begins by recalling the generators (e_{i}, f_{i}, k_{i}^{\pm1}) ((i=1,2)) and the four families of relations (A1–A4). Relations (A2) and (A3) describe the interaction between the Cartan‑like elements (k_{i}) and the root generators, while (A4) encodes a q‑Serre relation that is deformed by the sign parameters. A Hopf algebra structure is given by explicit coproduct, counit and antipode formulas (B1–B8). The authors also exhibit a non‑degenerate skew‑dual pairing between two Hopf subalgebras, showing that the whole algebra can be viewed as a Drinfeld double (D(\widehat B,\widehat B’)).

A crucial technical ingredient is the introduction of three families of automorphisms. The braid‑group‑type automorphisms (T_{i}) (Proposition 2.5) implement a quantum analogue of Weyl reflections; the index‑swapping automorphism (\Phi) exchanges the two simple roots; and the anti‑automorphism (\Omega) interchanges creation and annihilation operators while inverting the deformation parameter (q). These maps satisfy (\Omega^{2}= \Phi^{2}= \mathrm{id}) and intertwine in a controlled way, which is essential for constructing invariant objects later on.

Using (T_{i}) and (\Phi), the authors define real root vectors \